Sparse Matrices

Sparse matrices in Python#

The procedure we have used so far to construct the matrix for a is not very efficient. A full matrix is created, which consists mostly of zeros with non-zero values only appearing on diagonals. There are more efficient routines that store what are called sparse matrices. In a sparse matrix, only the value and location of non-zero values in a matrix are stored. Functionality for sparse matrices is available in the scipy submodule sparse.

We will import the scipy.sparse submodule as sp.

import scipy.sparse as sp

Dense to sparse#

When we create a sparse matrix, we have to choose which format it should be stored in. There are many ways to store a sparse matrix. Two most commonly used are:

1. compressed sparse column using the csc_matrix(A) method.
2. compressed sparse row using the csr_matrix(A) method.

The following matrix is an example of this:

Let’s convert this dense matrix to a sparse matrix. In the first example, we will convert it into a compressed sparse column:

In line 7, the sparse matrix is stored column-wise.

Now let’s convert it into a compressed sparse row:

In line 7, the sparse matrix is stored row-wise.

Sparse to dense#

todense() from the scipy.sparse submodule converts a sparse matrix to a dense matrix irrespective of its type. Let’s see an implementation of this below:

Solving equations with sparse matrices#

The solution to the system of equations $Ax=b$ is obtained with the spsolve() function of the scipy.sparse.linalg module.

Note: The multiplication sign does not do term-by-term multiplication for sparse matrix objects.

All the computations that we performed on dense matrices, can be performed on sparse matrices as well.

Let’s test your knowledge of this chapter with a quiz.